Multiplication Function on Ring with Unity is Zero if Characteristic is Divisor

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let the characteristic of $R$ be $p$.

Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:

$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$

where $\cdot$ denotes the multiple operation.


Then:

$p \divides n \implies n \cdot a = 0_R$

where $p \divides n$ denotes that $p$ is a divisor of $n$.


Proof

Let $p > 0$.

From Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function:

$\ideal p \subseteq \map \ker {g_a}$

where:

$\map \ker {g_a}$ is the kernel of $g_a$
$\ideal p$ is the principal ideal of $\Z$ generated by $p$.


We have:

\(\ds p\) \(\divides\) \(\ds n\)
\(\ds \leadsto \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds k p\) \(=\) \(\ds n\)


Then we have:

\(\ds kp = n\) \(\in\) \(\ds \ideal p\) Integral Ideal iff Set of Integer Multiples
\(\ds \leadsto \ \ \) \(\ds n\) \(\in\) \(\ds \map \ker {g_a}\) Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function
\(\ds \leadsto \ \ \) \(\ds \map {g_a} n\) \(=\) \(\ds 0\) Definition of Kernel of Group Homomorphism
\(\ds \leadsto \ \ \) \(\ds n \cdot a\) \(=\) \(\ds 0\) Definition of $g_a$

$\blacksquare$


Sources

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